arXiv:1001.1958 · Physical Review E

Lower bounds on the ground-state entropy of the Potts antiferromagnet on slabs of the simple cubic lattice

Robert E. Shrock, Yan Xu

Published Mar 31, 2010• 2 citations

Abstract

We calculate rigorous lower bounds for the ground-state degeneracy per site, W, of the q-state Potts antiferromagnet on slabs of the simple cubic lattice that are infinite in two directions and finite in the third and that thus interpolate between the square (sq) and simple cubic (sc) lattices. We give a comparison with large-q series expansions for the sq and sc lattices and also present numerical comparisons.

DOI: 10.1103/physreve.81.031134

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arXiv:1001.1958 · Physical Review E

Lower bounds on the ground-state entropy of the Potts antiferromagnet on slabs of the simple cubic lattice

Robert E. Shrock, Yan Xu

Published Mar 31, 2010• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreve.81.031134

Related papers

Large-q series expansion for the ground-state degeneracy of the q-state Potts antiferromagnet on the (3⋅12^{2}) lattice

Shan-Ho Tsai

We calculate the large- $q$ series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the $q$ -state Potts antiferromagnet on the $(3\ensuremath{\cdot}{12}^{2})$ lattice, to order ${O(y}^{19})$ , where $y=1/(q\ensuremath{-}1)$. We note a remarkable agreement, to ${O(y}^{18})$ , between this series and a rigorous lower bound derived recently.We calculate the large-$q$ series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the $q$-state Potts antiferromagnet on the $(3\ensuremath{\cdot}{12}^{2})$ lattice, to order ${O(y}^{19})$, where $y=1/(q\ensuremath{-}1)$. We note a remarkable agreement, to ${O(y}^{18})$, between this series and a rigorous lower bound derived recently.

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We calculate the large-

q

series expansion for the ground-state degeneracy (= exponent of the ground-state entropy) per site of the

q

-state Potts antiferromagnet on the $(3\ensuremath{\cdot}{12}^{2})$ lattice, to order

{O(y}^{19})

, where $y=1/(q\ensuremath{-}1)$. We note a remarkable agreement, to

{O(y}^{18})

, between this series and a rigorous lower bound derived recently.

We study the ground state degeneracy per site (exponent of the ground state entropy) $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ for the

q

-state Potts antiferromagnet on infinitely long strips with width

{L}_{y}

of 2D lattices \ensuremath{\Lambda} with free and periodic boundary conditions in the

y

direction, denoted

{\mathrm{FBC}}_{y}

and

{\mathrm{PBC}}_{y}.

We show that the approach of

W

to its 2D thermodynamic limit as

{L}_{y}

increases is quite rapid; for moderate values of

q

and ${L}_{y}\ensuremath{\simeq}4,$ $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{L}_{y},q)$ is within about 5% and ${O(10}^{\ensuremath{-}3})$ of the 2D value $W(\ensuremath{\Lambda}{,(L}_{x}=\ensuremath{\infty})\ifmmode\times\else\texttimes\fi{}{(L}_{y}=\ensuremath{\infty}),q)$ for

{\mathrm{FBC}}_{y}

and

{\mathrm{PBC}}_{y},

respectively. The approach of

W

to the 2D thermodynamic limit is proved to be monotonic (nonmonotonic) for

{\mathrm{FBC}}_{y}

({\mathrm{PBC}}_{y}).

It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states.

We present exact solutions for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial

P)

on tube sections of the simple cubic lattice of fixed transverse size ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ and arbitrarily great length

{L}_{z},

for sizes ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}=2\ifmmode\times\else\texttimes\fi{}3$ and $2\ifmmode\times\else\texttimes\fi{}4$ and boundary conditions (a)

({\mathrm{FBC}}_{x},{\mathrm{FBC}}_{y},{\mathrm{FBC}}_{z})

and (b)

({\mathrm{PBC}}_{x},{\mathrm{FBC}}_{y},{\mathrm{FBC}}_{z}),

where FBC (PBC) denote free (periodic) boundary conditions. In the limit of infinite length, ${L}_{z}\ensuremath{\rightarrow}\ensuremath{\infty},$ we calculate the resultant ground-state degeneracy per site W (=exponent of the ground-state entropy). Generalizing q from

{Z}_{+}

C,

we determine the analytic structure of W and the related singular locus

\mathcal{B}

which is the continuous accumulation set of zeros of the chromatic polynomial. For the ${L}_{z}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of a given family of lattice sections, W is analytic for real q down to a value

{q}_{c}.

We determine the values of

{q}_{c}

for the lattice sections considered and address the question of the value of

{q}_{c}

for a d-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph

{K}_{m,m}.